# The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular?

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The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular to the y-axis are equilateral triangles.

Find the volume V of this solid.

• Base of solid is region bound by line x+y=2, x-axis and y-axis.

Since cross-sections are perpendicular to y-axis, we will integrate with respect to y.

Function: x = 2 – y, limits: y=0 to y=2

Each cross-section is equilateral triangle with base = 2-y

Area of each cross-section = 1/2 * (2-y) * (2-y) * sin(60) = √(3)/4 (2-y)²

Integrating from 0 to 2, we get

V = √(3)/4 ∫₀² (2-y)² dy

V = √(3)/4 ∫₀² (4 – 4y + y²) dy

V = √(3)/4 (4y – 2y² + y³/3) |₀²

V = √(3)/4 [(8 – 8 + 8/3) – 0]

V = √(3)/4 * 8/3

V = 2√(3)/3

V = 1.1547

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